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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 560531, 8 pages
http://dx.doi.org/10.1155/2012/560531
Research Article

Traveling Wave Solutions of the Nonlinear (3+1)-Dimensional Kadomtsev-Petviashvili Equation Using the Two Variables (𝐺′/𝐺,1/𝐺)-Expansion Method

Mathematics Department, Faculty of Science, Zagazig University, Zagazig 44519, Egypt

Received 20 June 2012; Revised 30 July 2012; Accepted 31 July 2012

Academic Editor: TurgutΒ Γ–ziş

Copyright Β© 2012 E. M. E. Zayed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method is proposed in this paper to construct new exact traveling wave solutions with parameters of the nonlinear (3+1)-dimensional Kadomtsev-Petviashvili equation. This method can be considered as an extension of the basic (πΊξ…ž/𝐺)-expansion method obtained recently by Wang et al. When the parameters are replaced by special values, the well-known solitary wave solutions and the trigonometric periodic solutions of this equation were rediscovered from the traveling waves.

1. Introduction

In the recent years, investigations of exact solutions to nonlinear PDEs play an important role in the study of nonlinear physical phenomena. Many powerful methods have been presented, such as the inverse scattering transform method [1], the Hirota method [2], the truncated PainlevΓ© expansion method [3–6], the Backlund transform method [7, 8], the exp-function method [9–14], the tanh function method [15–18], the Jacobi elliptic function expansion method [19–21], the original (πΊξ…ž/𝐺)-expansion method [22–33], the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method [34, 35], and the first integral method [36]. The key idea of the original (πΊξ…ž/𝐺)-expansion method is that the exact solutions of nonlinear PDEs can be expressed by a polynomial in one variable (πΊξ…ž/𝐺) in which 𝐺=𝐺(πœ‰) satisfies the second ordinary differential equation πΊξ…žξ…ž(πœ‰)+πœ†πΊξ…ž(πœ‰)+πœ‡πΊ(πœ‰)=0, where πœ† and πœ‡ are constants. In this paper, we will use the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method, which can be considered as an extension of the original (πΊξ…ž/𝐺)-expansion method. The key idea of the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method is that the exact traveling wave solutions of nonlinear PDEs can be expressed by a polynomial in the two variables (πΊξ…ž/𝐺) and (1/𝐺), in which 𝐺=𝐺(πœ‰) satisfies a second order linear ODE, namely, πΊξ…žξ…ž(πœ‰)+πœ†πΊ(πœ‰)=πœ‡, where πœ† and πœ‡ are constants. The degree of this polynomial can be determined by considering the homogeneous balance between the highest order derivatives and nonlinear terms in the given nonlinear PDEs, while the coefficients of this polynomial can be obtained by solving a set of algebraic equations resulted from the process of using the method. According to Aslan [29], the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method becomes the basic (πΊξ…ž/𝐺)-expansion method if πœ‡=0 in (2.1) and 𝑏𝑖=0 in (2.12). Recently, Li et al. [34] have applied the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method and determined the exact solutions of the Zakharov equations, while Zayed and abdelaziz [35] have applied this method to determine the exact solutions of the nonlinear KdV-mKdV equation.

The objective of this paper is to apply the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method to find the exact traveling wave solutions of the following nonlinear (3+1)-dimensional Kadomtsev-Petviashvili equation: 𝑒π‘₯𝑑𝑒+6π‘₯ξ€Έ2+6𝑒𝑒π‘₯π‘₯βˆ’π‘’π‘₯π‘₯π‘₯π‘₯βˆ’π‘’π‘¦π‘¦βˆ’π‘’π‘§π‘§=0.(1.1) This equation describes the dynamics of solitons and nonlinear wave in plasma and superfluids. Recently, Zayed [24] has found the exact solutions of (1.1) using the original (πΊξ…ž/𝐺)-expansion method, while Aslan [14] has discussed (1.1) using the exp-function method. Comparison between our results and that obtained in [14, 24] will be discussed later. The rest of this paper is organized as follows. In Section 2, the description of the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method is given. In Section 3, we apply this method to (1.1). In Section 4, conclusions are obtained.

2. Description of the Two Variables (πΊξ…ž/𝐺,1/𝐺)-Expansion Method

Before we describe the main steps of this method, we need the following remarks (see [34, 35]).

Remark 2.1. If we consider the second order linear ODE πΊξ…žξ…ž(πœ‰)+πœ†πΊ(πœ‰)=πœ‡,(2.1) and set πœ™=πΊξ…ž/𝐺 and πœ“=1/𝐺, then we get πœ™ξ…ž=βˆ’πœ™2+πœ‡πœ“βˆ’πœ†,πœ“ξ…ž=βˆ’πœ™πœ“.(2.2)

Remark 2.2. If πœ†<0, then the general solutions of (2.1) is 𝐺(πœ‰)=𝐴1ξ‚€πœ‰βˆšsinhξ‚βˆ’πœ†+𝐴2ξ‚€πœ‰βˆšcosh+πœ‡βˆ’πœ†πœ†,(2.3) where 𝐴1 and 𝐴2 are arbitrary constants. Consequently, we have πœ“2=βˆ’πœ†πœ†2𝜎+πœ‡2ξ€·πœ™2ξ€Έ,βˆ’2πœ‡πœ“+πœ†(2.4) where 𝜎=𝐴21βˆ’π΄22.

Remark 2.3. If πœ†>0, then the general solutions of (2.1) is 𝐺(πœ‰)=𝐴1ξ‚€πœ‰βˆšsinπœ†ξ‚+𝐴2ξ‚€πœ‰βˆšcosπœ†ξ‚+πœ‡πœ†,(2.5) and hence πœ“2=βˆ’πœ†πœ†2πœŽβˆ’πœ‡2ξ€·πœ™2ξ€Έ.βˆ’2πœ‡πœ“+πœ†(2.6) where 𝜎=𝐴21+𝐴22.

Remark 2.4. If πœ†=0, then the general solutions of (2.1) is πœ‡πΊ(πœ‰)=2πœ‰2+𝐴1πœ‰+𝐴2,(2.7) and hence πœ“2=1𝐴21βˆ’2πœ‡π΄2ξ€·πœ™2ξ€Έ,βˆ’2πœ‡πœ“(2.8)

Suppose we have the following NLPDEs in the form: 𝐹𝑒,𝑒𝑑,𝑒π‘₯,𝑒π‘₯π‘₯,𝑒𝑑𝑑,…=0,(2.9) where 𝐹 is a polynomial in 𝑒 and its partial derivatives. In the following, we give the main steps of the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method [34, 35].

Step 1. The traveling wave variable 𝑒(π‘₯,𝑑)=𝑒(πœ‰),πœ‰=π‘₯βˆ’π‘‰π‘‘(2.10) reduces (2.9) to an ODE in the form 𝑃𝑒,π‘’ξ…ž,π‘’ξ…žξ…žξ€Έ,…=0,(2.11) where 𝑉 is a constant and 𝑃 is a polynomial in 𝑒 and its total derivatives, while {}ξ…ž=𝑑/π‘‘πœ‰.

Step 2. Suppose that the solutions of (2.11) can be expressed by a polynomial in the two variables πœ™ and πœ“ as follows: 𝑒(πœ‰)=𝑖=𝑁𝑖=0π‘Žπ‘–πœ™π‘–+𝑖=𝑁𝑖=1π‘π‘–πœ™π‘–βˆ’1πœ“,(2.12) where π‘Žπ‘–(𝑖=0,1,…,𝑁) and 𝑏𝑖(𝑖=1,…,𝑁) are constants to be determined later.

Step 3. Determine the positive integer 𝑁 in (2.12) by using the homogeneous balance between the highest order derivatives and the nonlinear terms in (2.11).

Step 4. Substitute (2.12) into (2.11) along with (2.2) and (2.4), the left-hand side of (2.11) can be converted into a polynomial in πœ™ and πœ“, in which the degree of πœ“ is not longer than 1. Equating each coefficients of this polynomial to zero yields a system of algebraic equations which can be solved by using the Maple or Mathematica to get the values of π‘Žπ‘–,𝑏𝑖,𝑉,πœ‡,𝐴1,𝐴2, and πœ† where πœ†<0. Thus, we get the exact solutions in terms of the hyperbolic functions.

Step 5. Similar to Step 4, substitute (2.12) into (2.11) along with (2.2) and (2.6) for πœ†>0 (or (2.2) and (2.8) for πœ†=0), we obtain the exact solutions of (2.11) expressed by trigonometric functions (or by rational functions), respectively.

3. An Application

In this section, we apply the method described in Section 2, to find the exact traveling wave solutions of the nonlinear (3+1)-dimensional Kadomtsev-Petviashvili equation (1.1). To this end, we see that the traveling wave variables πœ‰=π‘₯+𝑦+π‘§βˆ’π‘‰π‘‘ reduce (1.1) to the following ODE: βˆ’(2+𝑉)π‘’ξ…žξ…žξ€·π‘’+6ξ…žξ€Έ2+6π‘’π‘’ξ…žξ…žβˆ’π‘’ξ…žξ…žξ…žξ…ž=0.(3.1) Balancing π‘’ξ…žξ…žξ…žξ…ž with π‘’π‘’ξ…žξ…ž in (3.1) we get 𝑁=2. Consequently, we get 𝑒(πœ‰)=π‘Ž0+π‘Ž1πœ™(πœ‰)+π‘Ž2πœ™2(πœ‰)+𝑏1πœ“(πœ‰)+𝑏2πœ™(πœ‰)πœ“(πœ‰),(3.2) where π‘Ž0,π‘Ž1,π‘Ž2,𝑏1, and 𝑏2 are constants to be determined later. There are three cases to be discussed as follows.

Case 1. Hyperbolic function solutions (πœ†<0).
If πœ†<0, substituting (3.2) into (3.1) and using (2.2) and (2.4), the left-hand side of (3.1) becomes a polynomial in πœ™ and πœ“. Setting the coefficients of this polynomial to zero yields a system of algebraic equations in π‘Ž0,π‘Ž1,π‘Ž2,𝑏1,𝑏2,πœ‡,𝜎, and πœ† which can be solved by using the Maple or Mathematica to find the following results: π‘Ž0=π‘Ž0,π‘Ž1=0,π‘Ž2=1,𝑏1=βˆ’πœ‡,𝑏2ξƒŽ=Β±βˆ’ξ€·πœ†2𝜎+πœ‡2ξ€Έπœ†,𝑉=6π‘Ž0βˆ’5πœ†βˆ’2.(3.3)
From (2.3), (3.2), and (3.3), we deduce the traveling wave solution of (1.1) as follows: 𝑒(πœ‰)=π‘Ž0βˆ’πœ‡π΄1ξ‚€πœ‰βˆšsinhξ‚βˆ’πœ†+𝐴2ξ‚€πœ‰βˆšcosh+βˆ’π΄βˆ’πœ†(πœ‡/πœ†)1ξ‚€πœ‰βˆšcoshξ‚βˆ’πœ†+𝐴2ξ‚€πœ‰βˆšsinhξ‚βˆ’πœ†ξ‚€π΄1ξ‚€πœ‰βˆšsinhξ‚βˆ’πœ†+𝐴2ξ‚€πœ‰βˆšcoshξ‚ξ‚βˆ’πœ†+πœ‡/πœ†2×𝐴1ξ‚€πœ‰βˆšπœ†coshξ‚βˆ’πœ†+𝐴2ξ‚€πœ‰βˆšπœ†sinhξ‚βˆ“ξ”βˆ’πœ†πœ†2𝜎+πœ‡2ξ‚Ή,(3.4) where ξ€·πœ‰=π‘₯+𝑦+π‘§βˆ’6π‘Ž0ξ€Έβˆ’5πœ†βˆ’2𝑑.(3.5) In particular, by setting 𝐴1=0,𝐴2>0 and πœ‡=0 in (3.4), we have the solitary solution 𝑒(πœ‰)=π‘Ž0ξ‚€πœ‰βˆšβˆ’πœ†tanhξ‚€πœ‰βˆšβˆ’πœ†ξ‚ξ‚ƒtanhξ‚βˆ’πœ†βˆ“π‘–sechξ‚€πœ‰βˆš,βˆ’πœ†ξ‚ξ‚„(3.6) where βˆšπ‘–=βˆ’1, while if 𝐴2=0,𝐴1>0, and πœ‡=0, then we have the solitary solution 𝑒(πœ‰)=π‘Ž0ξ‚€πœ‰βˆšβˆ’πœ†cothξ‚€πœ‰βˆšβˆ’πœ†ξ‚ξ‚ƒcothξ‚βˆ“βˆ’πœ†cschξ‚€πœ‰βˆš.βˆ’πœ†ξ‚ξ‚„(3.7)

Case 2. Trigonometric function solutions (πœ†>0).
If πœ†>0, substituting (3.2) into (3.1) and using (2.2) and (2.6), we get a polynomial in πœ™ and πœ“. Vanishing each coefficient of this polynomial to get the algebraic equations which can be solved by using the Maple or Mathematica to find the following results: π‘Ž0=π‘Ž0,π‘Ž1=0,π‘Ž2=1,𝑏1=βˆ’πœ‡,𝑏2ξƒŽ=Β±πœ†2πœŽβˆ’πœ‡2πœ†,𝑉=6π‘Ž0βˆ’5πœ†βˆ’2.(3.8)
From (2.5), (3.2), and (3.8), we deduce the traveling wave solution of (1.1) as follows: 𝑒(πœ‰)=π‘Ž0βˆ’πœ‡π΄1ξ‚€πœ‰βˆšsinπœ†ξ‚+𝐴2ξ‚€πœ‰βˆšcosπœ†ξ‚++𝐴(πœ‡/πœ†)1ξ‚€πœ‰βˆšcosπœ†ξ‚βˆ’π΄2ξ‚€πœ‰βˆšsinπœ†ξ‚ξ‚€π΄1ξ‚€πœ‰βˆšsinπœ†ξ‚+𝐴2ξ‚€πœ‰βˆšcosπœ†ξ‚ξ‚+πœ‡/πœ†2×𝐴1ξ‚€πœ‰βˆšπœ†cosπœ†ξ‚βˆ’π΄2ξ‚€πœ‰βˆšπœ†sinπœ†ξ‚Β±ξ”πœ†2πœŽβˆ’πœ‡2ξ‚Ή,(3.9) where πœ‰ has the same form (3.5).
In particular, by setting 𝐴1=0,𝐴2>0, and πœ‡=0 in (3.9), we have the periodic solution 𝑒(πœ‰)=π‘Ž0ξ‚€πœ‰βˆš+πœ†tanπœ†ξ‚€πœ‰βˆšξ‚ξ‚ƒtanπœ†ξ‚βˆ“secξ‚€πœ‰βˆšπœ†,(3.10) while if 𝐴2=0,𝐴1>0, and πœ‡=0, then we have the periodic solution 𝑒(πœ‰)=π‘Ž0+πœ†cotξ‚€πœ‰βˆšπœ†ξ‚ξ‚ƒcotξ‚€πœ‰βˆšπœ†ξ‚Β±cscξ‚€πœ‰βˆšπœ†.(3.11)

Case 3. Rational function solutions (πœ†=0).
If πœ†=0, substituting (3.2) into (3.1) and using (2.2) and (2.8), we get a polynomial in πœ™ and πœ“. Setting each coefficients of this polynomial to be zero to get the algebraic equations which can be solved by using the Maple or Mathematica to find the following results: π‘Ž0=π‘Ž0,π‘Ž1=0,π‘Ž2=1,𝑏1=βˆ’πœ‡,𝑏2=±𝐴21βˆ’2πœ‡π΄2,𝑉=6π‘Ž0βˆ’2.(3.12) From (2.7), (3.2), and (3.12), we deduce the traveling wave solution of (1.1) as follows: 𝑒(πœ‰)=π‘Ž0βˆ’πœ‡(πœ‡/2)πœ‰2+𝐴1πœ‰+𝐴2+ξ€·πœ‡πœ‰+𝐴1ξ€Έξ‚΅πœ‡πœ‰+𝐴1±𝐴21βˆ’2πœ‡π΄2ξ‚Άξ€·(πœ‡/2)πœ‰2+𝐴1πœ‰+𝐴2ξ€Έ2,(3.13) where ξ€·πœ‰=π‘₯+𝑦+π‘§βˆ’6π‘Ž0ξ€Έβˆ’2𝑑.(3.14)

Remark 3.1. All solutions of this paper have been checked with Maple by putting them back into the original equation (1.1).

4. Conclusions

The two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method has been used in this paper to discuss (1.1) and obtain the exact traveling wave solutions (3.4), (3.9), and (3.13) of Section 3. As the two parameters 𝐴1 and 𝐴2 take special values, we obtain the solitary wave solutions (3.6) and (3.7) and the trigonometric periodic solutions (3.10) and (3.11). On comparing these solutions with the result (11) of [14] obtained by Aslan using the exp-function method as well as the results (3.28)–(3.31) of [24] obtained by Zayed using the basic (πΊξ…ž/𝐺)-expansion method, we conclude that all these solutions of (1.1) are different and satisfying that equation. The advantage of the two variables (πΊξ…ž/𝐺,1/𝐺)-expansion method over the basic (πΊξ…ž/𝐺)-expansion method is that the first method is an extension of the second one.

Acknowledgment

The authors wish to thank the referee for his suggestions and comments.

References

  1. M. J. Ablowitz and P. A. Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering Transform, Cambridge University Press, New York, NY, USA, 1991. View at Publisher Β· View at Google Scholar
  2. R. Hirota, β€œExact solutions of the KdV equation for multiple collisions of solutions,” Physical Review Letters, vol. 27, pp. 1192–1194, 1971. View at Google Scholar
  3. J. Weiss, M. Tabor, and G. Carnevale, β€œThe Painlevé property for partial differential equations,” Journal of Mathematical Physics, vol. 24, no. 3, pp. 522–526, 1983. View at Publisher Β· View at Google Scholar
  4. N. A. Kudryashov, β€œExact soliton solutions of a generalized evolution equation of wave dynamics,” Journal of Applied Mathematics and Mechanics, vol. 52, pp. 361–365, 1988. View at Google Scholar
  5. N. A. Kudryashov, β€œExact solutions of the generalized Kuramoto-Sivashinsky equation,” Physics Letters A, vol. 147, no. 5-6, pp. 287–291, 1990. View at Publisher Β· View at Google Scholar
  6. N. A. Kudryashov, β€œOn types of nonlinear nonintegrable equations with exact solutions,” Physics Letters A, vol. 155, no. 4-5, pp. 269–275, 1991. View at Publisher Β· View at Google Scholar
  7. M. R. Miura, Bäcklund Transformation, Springer, Berlin, Germany, 1978.
  8. C. Rogers and W. F. Shadwick, Bäcklund Transformations and Their Applications, vol. 161, Academic Press, New York, NY, USA, 1982.
  9. J.-H. He and X.-H. Wu, β€œExp-function method for nonlinear wave equations,” Chaos, Solitons and Fractals, vol. 30, no. 3, pp. 700–708, 2006. View at Publisher Β· View at Google Scholar
  10. E. Yusufoglu, β€œNew solitary for the MBBM equations using Exp-function method,” Physics Letters A, vol. 372, pp. 442–446, 2008. View at Google Scholar
  11. S. Zhang, β€œApplication of Exp-function method to high-dimensional nonlinear evolution equation,” Chaos, Solitons and Fractals, vol. 38, no. 1, pp. 270–276, 2008. View at Publisher Β· View at Google Scholar
  12. A. Bekir, β€œThe exp-function for Ostrovsky equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, pp. 735–739, 2009. View at Google Scholar
  13. A. Bekir, β€œApplication of the exp-function method for nonlinear differential-difference equations,” Applied Mathematics and Computation, vol. 215, no. 11, pp. 4049–4053, 2010. View at Publisher Β· View at Google Scholar
  14. I. Aslan, β€œComment on: “Application of Exp-function method for (3+1 )-dimensional nonlinear evolution equations” [Comput. Math. Appl. 56 (2008) 1451–1456],” Computers and Mathematics with Applications, vol. 61, no. 6, pp. 1700–1703, 2011. View at Publisher Β· View at Google Scholar Β· View at Scopus
  15. M. A. Abdou, β€œThe extended tanh method and its applications for solving nonlinear physical models,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 988–996, 2007. View at Publisher Β· View at Google Scholar
  16. E. Fan, β€œExtended tanh-function method and its applications to nonlinear equations,” Physics Letters A, vol. 277, no. 4-5, pp. 212–218, 2000. View at Publisher Β· View at Google Scholar
  17. S. Zhang and T.-c. Xia, β€œA further improved tanh function method exactly solving the (2+1)-dimensional dispersive long wave equations,” Applied Mathematics E-Notes, vol. 8, pp. 58–66, 2008. View at Google Scholar
  18. E. Yusufoğlu and A. Bekir, β€œExact solutions of coupled nonlinear Klein-Gordon equations,” Mathematical and Computer Modelling, vol. 48, no. 11-12, pp. 1694–1700, 2008. View at Publisher Β· View at Google Scholar
  19. Y. Chen and Q. Wang, β€œExtended Jacobi elliptic function rational expansion method and abundant families of Jacobi elliptic function solutions to (1+1)-dimensional dispersive long wave equation,” Chaos, Solitons and Fractals, vol. 24, no. 3, pp. 745–757, 2005. View at Publisher Β· View at Google Scholar
  20. S. Liu, Z. Fu, S. Liu, and Q. Zhao, β€œJacobi elliptic function expansion method and periodic wave solutions of nonlinear wave equations,” Physics Letters A, vol. 289, no. 1-2, pp. 69–74, 2001. View at Publisher Β· View at Google Scholar
  21. D. Lü, β€œJacobi elliptic function solutions for two variant Boussinesq equations,” Chaos, Solitons and Fractals, vol. 24, no. 5, pp. 1373–1385, 2005. View at Publisher Β· View at Google Scholar
  22. M. Wang, X. Li, and J. Zhang, β€œThe (G/G)-expansion method and travelling wave solutions of nonlinear evolution equations in mathematical physics,” Physics Letters A, vol. 372, no. 4, pp. 417–423, 2008. View at Publisher Β· View at Google Scholar
  23. S. Zhang, J. L. Tong, and W. Wang, β€œA generalized (G/G)-expansion method for the mKdV equation with variable coefficients,” Physics Letters A, vol. 372, pp. 2254–2257, 2008. View at Google Scholar
  24. E. M. E. Zayed, β€œTraveling wave solutions for higher-dimensional nonlinear evolution equations using the (G/G)-expansion method,” Journal of Applied Mathematics & Informatics, vol. 28, pp. 383–395, 2010. View at Google Scholar
  25. E. M. E. Zayed, β€œThe (G/G)-expansion method and its applications to some nonlinear evolution equations in the mathematical physics,” Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 89–103, 2009. View at Publisher Β· View at Google Scholar
  26. A. Bekir, β€œApplication of the (G/G)-expansion method for nonlinear evolution equations,” Physics Letters A, vol. 372, no. 19, pp. 3400–3406, 2008. View at Publisher Β· View at Google Scholar
  27. B. Ayhan and A. Bekir, β€œThe (G/G)-expansion method for the nonlinear lattice equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 9, pp. 3490–3498, 2012. View at Google Scholar
  28. N. A. Kudryashov, β€œA note on the (G/G)-expansion method,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1755–1758, 2010. View at Publisher Β· View at Google Scholar
  29. I. Aslan, β€œA note on the (G/G)-expansion method again,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 937–938, 2010. View at Publisher Β· View at Google Scholar
  30. N. A. Kudryashov, β€œMeromorphic solutions of nonlinear ordinary differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 10, pp. 2778–2790, 2010. View at Publisher Β· View at Google Scholar
  31. I. Aslan, β€œExact and explicit solutions to the discrete nonlinear Schrödinger equation with a saturable nonlinearity,” Physics Letters A, vol. 375, no. 47, pp. 4214–4217, 2011. View at Publisher Β· View at Google Scholar Β· View at Scopus
  32. I. Aslan, β€œSome exact solutions for Toda type lattice differential equations using the improved (G/G)-expansion method,” Mathematical Methods in the Applied Sciences, vol. 35, no. 4, pp. 474–481, 2012. View at Publisher Β· View at Google Scholar Β· View at Scopus
  33. I. Aslan, β€œThe discrete (G/G)-expansion method applied to the differential-difference Burgers equation and the relativistic Toda lattice system,” Numerical Methods for Partial Differential Equations. An International Journal, vol. 28, no. 1, pp. 127–137, 2012. View at Publisher Β· View at Google Scholar
  34. L.-x. Li, E.-q. Li, and M.-l. Wang, β€œThe (G/G,1/G)-expansion method and its application to travelling wave solutions of the Zakharov equations,” Applied Mathematics B, vol. 25, no. 4, pp. 454–462, 2010. View at Publisher Β· View at Google Scholar
  35. E. M. E. Zayed and M. A. M. Abdelaziz, β€œThe two variables (G/G,1/G) -expansion method for solving the nonlinear KdV-mKdV equation,” Mathematical Problems in Engineering, vol. 2012, Article ID 725061, 14 pages, 2012. View at Publisher Β· View at Google Scholar
  36. F. Tascan and A. Bekir, β€œApplications of the first integral method to the nonlinear evolution equations,” Chinese Physics B, vol. 19, Article ID 080201, 11 pages, 2010. View at Google Scholar